The purpose of this post is to serve as a reference on standard properties of the driven, damped harmonic oscillator:
\[\ddot{x}+\Delta\omega\dot{x}+\omega_0^2x=f(t)\]
where the damping coefficient \(\Delta\omega>0\) describes the resonant bandwidth of the system’s frequency response (reciprocal to the oscillator lifetime \(\tau=1/\Delta\omega\)), and \(\omega_0>0\) is the natural (and approximately resonant) angular frequency of the harmonic oscillator. The ratio of these \(2\) frequency parameters yields a dimensionless quality factor of the damped harmonic oscillator:
\[Q:=\frac{\omega_0}{\Delta\omega}\]
where a higher \(Q\)-factor, as its name suggests, indicates a less damped oscillator, hence one that “rings” for longer before it dies out.
The general solution of the inhomogeneous/driven ODE will look like an exponentially decaying transient complementary function that solves the homogeneous/undriven ODE (dependent on initial conditions), superimposed with a steady state particular integral that oscillates at the driving frequency of \(f(t)\) (independent of initial conditions).
Henceforth focusing on the steady state, if \(f(t)=f_0e^{i\omega t}\) (and one can gauge-fix \(f_0\in (0,\infty)\)), then the ansatz \(x(t)=x_0e^{i\omega t}\) yields the frequency response:
\[x_0=\frac{f_0}{\omega_0^2-\omega^2+i\omega\Delta\omega}\]
The point is to quickly extract amplitude and phase information about the response:
\[|x_0|=\frac{f_0}{\sqrt{(\omega^2-\omega^2_0)^2+\omega^2\Delta\omega^2}}\]
\[\tan\angle x_0=\frac{\omega\Delta\omega}{\omega^2-\omega_0^2}\]
Although the response amplitude \(|x_0|\) is strictly maximized when driven at \(\omega=\sqrt{\omega_0^2+\Delta\omega^2/2}\), in the weakly damped limit \(\Delta\omega\ll\omega_0\) this is \(\omega=\omega_0+O(\Delta\omega)^2\). At this frequency, the phase offset \(\angle x_0\) between \(f(t)\) and \(x(t)\) rapidly (i.e. over a “Fermi surface” of thickness \(\sim\Delta\omega\)) “flips” from being in-phase \(\angle x_0\approx 0\) to out-of-phase \(\angle x_0\approx\pi\).
In addition to \(x_0=x_0(\omega)\), another useful spectrum to look at is the steady state period-averaged driving power \(\bar P=\bar P(\omega)\) (note this is not the total power, which would also include the powers developed by the spring and damping forces, this is just the input driving power). First, the steady state driving power not time-averaged yet (for mass \(m\)) is:
\[P(t)=\Re mf(t)\Re \dot x(t)=mf_0\cos\omega t\Re\left(i\omega x_0e^{i\omega t}\right)\]
\[=\frac{m\omega f_0^2\cos \omega t}{(\omega^2-\omega_0^2)^2+\omega^2\Delta\omega^2}\left[(\omega^2-\omega_0^2)\sin\omega t+\omega\Delta\omega\cos\omega t\right]\]
so time-averaging over a period:
\[\bar P=\frac{m\omega^2 f_0^2\Delta\omega}{2((\omega^2-\omega_0^2)^2+\omega^2\Delta\omega^2)}\]
(alternatively, one could first compute the complex power \((mf)^{\dagger}\dot x/2\) and take \(\Re\) to obtain the real power injected into the system by driving, leading to the same result \(\bar P\) as above).
Working around resonance \(\omega\approx\omega_0\), the difference of squares approximation \(\omega^2-\omega_0^2\approx 2\omega_0(\omega-\omega_0)\) (and similar approximations) lead to:
\[|x_0|\approx\frac{f_0}{2\omega_0\sqrt{(\omega-\omega_0)^2+\Delta\omega^2/4}}\]
\[\tan\angle x_0=\frac{\Delta\omega}{2(\omega-\omega_0)}\]
\[\bar P=\frac{mf_0^2}{2\Delta\omega}\frac{1}{1+(2(\omega-\omega_0)/\Delta\omega)^2}\]
where the form of \(\bar P\) is immediately recognized as a Lorentzian with FWHM bandwidth \(\Delta\omega\).
A physically insightful way to remember this is as follows: in the steady state, the input driving power is matched by the dissipative power due to damping, so looking at the equation of motion, one can equate on dimensional grounds \(\dot x\Delta\omega\sim f_0\). But the power at resonance is \(F_0\dot x\sim F_0f_0/\Delta\omega\sim F_0^2/m\Delta\omega\). Then, just remember there’s a factor of \(1/2\) from period-averaging a \(\cos^2\).
Problem: Write down the general (i.e. initial condition dependent) transient particular integral solution to the homogeneous ODE (i.e. an undriven oscillator).
Solution: In the underdamped case \(Q>1/2\) (i.e. there will be oscillations if released from rest):
\[x(t)=e^{-\Delta\omega t/2}\left(A\cos\left(\sqrt{\omega_0^2-\frac{\Delta\omega^2}{4}}t\right)+B\sin\left(\sqrt{\omega_0^2-\frac{\Delta\omega^2}{4}}t\right)\right)\]
In the overdamped case \(Q<1/2\) (i.e. no oscillations if released from rest):
\[x(t)=e^{-\Delta\omega t/2}\left(Ae^{-\sqrt{\Delta\omega^2/4-\omega_0^2}t}+Be^{\sqrt{\Delta\omega^2/4-\omega_0^2}t}\right)\]
And in the critically damped case \(Q=1/2\):
\[x(t)=e^{-\Delta\omega t/2}(A+Bt)\]
All these are quickly checked by noting the identity:
\[\ddot x+\Delta\omega\dot x+\omega_0^2x=e^{-\Delta\omega t/2}\frac{d^2}{dt^2}\left(e^{\Delta\omega t/2}x\right)+\left(\omega_0^2-\frac{\Delta\omega^2}{4}\right)x\]
Misconception: An underdamped, undriven harmonic oscillator will have oscillations but a critically damped or underdamped (undriven) harmonic oscillator will not have oscillations.
Clarification: It is true that an underdamped oscillator (regardless of initial conditions) always oscillates before it dies out to \(x\to 0\). In the case of critical damping or being underdamped, if the oscillator starts from rest \(\dot{x}(0)=0\), then indeed it will not oscillate about \(x=0\) in the sense that there will be no overshoot, just a monotonic approach towards \(x=0\) with critical damping being the fastest. However, if one does give the system an initial push in the direction of \(x=0\), then indeed the oscillator can (though not necessarily! One requires \(\dot{x}(0)<x(0)\Delta\omega/2\) for critical damping) overshoot the origin \(x=0\), though only once (if one pushes it in the direction away from \(x=0\), then indeed there will be no oscillation since that’s equivalent to the system just starting from a new amplitude and decaying monotonically back to \(x\to 0\)).
Misconception: The quality factor \(Q\) is only defined for an undriven oscillator.
Clarification: From the definition \(Q=\omega_0/\Delta\omega\), it should be clear that the quality factor is an intrinsic property of any oscillator, and whether or not it happens to be driven is immaterial to \(Q\) being well-defined. Nevertheless, the interpretation of \(Q\) is different depending on whether the oscillator is being driven or not:
- If the oscillator is undriven, then of course no matter whether it is underdamped, critically damped, or overdamped, one has \(\lim_{t\to\infty}x(t)=0\) simply because damping \(\Delta\omega>0\) exists. So in this case, the interpretation of \(Q\) is that it is the number of radians of oscillations before the energy of the oscillator dissipates to \(1/e\) of its initial energy. Equivalently, \(Q/2\pi\) is the number of oscillations before the oscillator “stops oscillating” and so this occurs after a time:
\[t=\frac{Q}{2\pi}T=\frac{1}{\Delta\omega}\]
cf. the exponentially decaying envelope \(x(t)\sim e^{-\Delta\omega t/2}\) in all cases of underdamped/critically damped/overdamped behavior. This interpretation is in fact best suited for underdamped oscillators since only they basically oscillate (though see the misconception above).
2. If the oscillator is driven, then \(Q\) is interpreted as the sharpness of the resonance peak of the response function \(x_0(\omega)\) as a function of the driving frequency \(\omega\).